This completes the matching of the most general EFT Lagrangian for
fermionic DM coupling to SM through operators up to and including
dimension seven. The cross section for DM scattering on nuclei, due to
DM scattering on a single nucleon, is given by Eqs. (40) and (46) of
Ref. [1]}. These expressions do receive large
corrections from the 2-nucleon interactions in the case of Rayleigh
operators. They are of the size \(\sim Z Q_0/m_N\)  relative to the
single-nucleon scattering, and can be dominant for large nuclei. Here
\(Q_0\sim 1/(r_0 A^{1/3})\sim 160~{\rm MeV}/A^{1/3}\)  is the momentum
scale corresponding to the size of the nucleus, see
Ref. [2]} for further details. For all the other
UV operators the two-nucleon contributions are subleading, see, e.g.,
Ref. [3]}, and the above expressions can be readily
used to obtain the leading-order expressions in chiral counting for
the DM–nucleus scattering cross section.
